A370446 Expansion of g.f. A(x) satisfying A(x)^3 + x^4/A(x)^3 = A(x^3) + x^4/A(x^3) - 3*x^2.

1, 1, 2, 6, 20, 71, 267, 1041, 4168, 17047, 70902, 298967, 1275141, 5491504, 23846271, 104295430, 459023543, 2031459236, 9034769573, 40358643042, 180998556943, 814645632727, 3678542796070, 16659932961647, 75657738747396, 344446195875766, 1571786529601990, 7187790264787872

Offset: 1

Author

Paul D. Hanna, Mar 09 2024

Keywords

• nonn

Examples

			G.f.: A(x) = x + x^2 + 2*x^3 + 6*x^4 + 20*x^5 + 71*x^6 + 267*x^7 + 1041*x^8 + 4168*x^9 + 17047*x^10 + 70902*x^11 + 298967*x^12 + 1275141*x^13 + 5491504*x^14 + 23846271*x^15 + ...
RELATED SERIES.
We can illustrate the formulas with the following related expansions.
(1) A(x)^3 + 2*x^2 + x^4/A(x)^3 = x - x^2 + x^3 - x^4 + x^6 - x^7 + 2*x^9 - 3*x^10 + 6*x^12 - 9*x^13 + 20*x^15 - 30*x^16 + 71*x^18 - 110*x^19 + 267*x^21 + ...
which equals A(x^3) - x^2 + x^4/A(x^3), as can be seen from
x^4/A(x^3) = x - x^4 - x^7 - 3*x^10 - 9*x^13 - 30*x^16 - 110*x^19 + ...
A(x)^3 = x^3 + 3*x^4 + 9*x^5 + 31*x^6 + 114*x^7 + 435*x^8 + 1715*x^9 + ...
x^4/A(x)^3 = x - 3*x^2 - 4*x^4 - 9*x^5 - 30*x^6 - 115*x^7 - 435*x^8 - 1713*x^9 + ...
(2) Let F(x) be the g.f. of A370440, which begins
F(x) = x + x^2 + x^3 + x^4 + 2*x^5 + 6*x^6 + 15*x^7 + 30*x^8 + 55*x^9 + 113*x^10 + 274*x^11 + 683*x^12 + ...
where F(x)^3 = F( x^3 + 3*x^2*F(x)^2 ),
then the series reversion of F(x) begins
A(x^3) - x^2 + x^4/A(x^3) = x - x^2 + x^3 - x^4 + x^6 - x^7 + 2*x^9 - 3*x^10 + 6*x^12 - 9*x^13 + 20*x^15 - 30*x^16 + 71*x^18 - 110*x^19 + 267*x^21 - 419*x^22 + 1041*x^24 + ...
(3) Let G(x) be the g.f. of A264228, which begins
G(x) = x + x^2 + 2*x^3 + 5*x^4 + 13*x^5 + 35*x^6 + 97*x^7 + 274*x^8 + 785*x^9 + 2275*x^10 + 6656*x^11 + 19630*x^12 +...
where G(x)^3 = G( x^3/(1 - 3*x) ),
then the series reversion of G(x) begins
-x^2/(A(-x^3) - x^2 + x^4/A(-x^3)) = x^2/(x + x^2 + x^3 + x^4 - x^6 - x^7 + 2*x^9 + 3*x^10 - 6*x^12 - 9*x^13 + 20*x^15 + 30*x^16 - 71*x^18 - 110*x^19 + 267*x^21 + 419*x^22 - 1041*x^24 +...).
SPECIFIC VALUES.
A(1/4.834464) = 0.349644497578571280258023712232522068793519739...
A(1/5) = 0.29940801195429552263938628184744484915469836164855...
A(1/6) = 0.21539123666426270273178791857213676628593723946879...
A(1/7) = 0.17414937372444126736977770687571455113383911571251...
A(1/8) = 0.14713126344900776621336355426627444003268957268553...
A(1/5^3) = 0.00806504925055020701973761348380106375185943151538...
A(1/6^3) = 0.00465126435780731657600811126033650347236250831668...
A(1/7^3) = 0.00292400175440295890949208907819991271975334925594...
which may be used to verify that the formula
A(x)^3 + x^4/A(x)^3 = A(x^3) + x^4/A(x^3) - 3*x^2
holds for these specific values.
		

Links

• Paul D. Hanna, Table of n, a(n) for n = 1..800

Crossrefs

Cf. A370440, A264228.

Programs

PARI

{a(n) = my(A=x); for(m=1,n, A=truncate(A) +x^4*O(x^m); A = ( x^4/(x^4/subst(A,x,x^3) + subst(A,x,x^3) - A^3 - 3*x^2) +x^4*O(x^n))^(1/3) );polcoeff(A,n)}
for(n=1,30,print1(a(n),", "))

Formula

G.f. A(x) = Sum_{n>=1} a(n)*x^n satisfies the following formulas.

(1) A(x)^3 + 2*x^2 + x^4/A(x)^3 = A(x^3) - x^2 + x^4/A(x^3).

(2) F( A(x^3) - x^2 + x^4/A(x^3) ) = x, where F(x) = F( x^3 + 3*x^2*F(x)^2 )^(1/3) is the g.f. of A370440.

(3) G( -x^2/(A(-x^3) - x^2 + x^4/A(-x^3)) ) = x, where G(x) = G( x^3/(1 - 3*x) )^(1/3) is the g.f. of A264228.

a(n) ~ c * d^n / n^(3/2), where d = 4.8344630246454026903035642546835542141482126303313357979263... and c = 0.0713578385738499677445741870058758452888939567284935382... - Vaclav Kotesovec, Mar 13 2024

The radius of convergence r = 0.20684820525095397... = 1/d (where d is given above), and A(r) = 0.3497581458819115559285308998459940399916633464611700768... satisfy A(r) = r^(2/3) and A(r^3) = (5 - sqrt(21))/2 * r^2. - Paul D. Hanna, Mar 13 2024